Toponia at the HL-LHC and FCC-ee

Abstract

The hint of a pseudoscalar toponium state at the Large Hadron Collider (LHC) opens a new avenue for studying a novel class of QCD (quasi-)bound states with comparable formation and decay times. Compared with charmonium and bottomonium, toponium is a quasi-bound state, resembling a hydrogen atom of the strong interaction, although it appears as a broader resonance. We compute the masses and annihilation decay widths of the lowest $S$-wave ($η_t$, $ψ_t$) and $P$-wave ($χ_{t0}$, $χ_{t1}$) toponium states, and assess their discovery prospects at the High-Luminosity LHC (HL-LHC) and future lepton colliders, such as the $e^+e^-$ stage of the Future Circular Collider (FCC-ee). Detecting the vector $ψ_t$ state at the HL-LHC is hindered by the Landau-Yang theorem and the gluon-dominated production environment of the collider, whereas lepton colliders offer promising sensitivity through both constituent and two-body decays. A more precise measurement of the $η_t$ mass, approximately equal to that of $ψ_t$, at the LHC could help determine the optimal $t\bar{t}$ threshold center-of-mass energy for FCC-ee. The $P$-wave states remain challenging to observe at both the HL-LHC and future lepton colliders. We also discuss how toponium measurements can be used to probe top-quark properties and to conduct indirect searches for new physics, including light scalars that couple to the top quark.

Figures (9)

• Figure 1: The mass spectrum of the lowest $S$- and $P$-wave toponium states, categorized by their principal quantum number $n$ and $n-1$ and $J^{\rm PC}$ quantum numbers. In this plot, we fix $m_t=172.4$ GeV, and the gray dashed line denotes the threshold mass $2m_t=344.8$ GeV, above and below which $E_{\rm bind}>0$ and $E_{\rm bind}<0$, respectively. The renormalization scale for $\alpha_s(\mu_R)$ in the static potential is chosen to be $\mu_R = 1/r$. There is an uncertainty of around $\delta M\sim 2$ GeV for all toponium masses resulting from the top quark mass and renormalization scale uncertainties. The mass difference between $\eta_t(n\hbox{S})$ and $\psi_t(n\hbox{S})$ is negligible and that among the $P$-wave states with the same principle quantum number $n-1$ is around $5~{\rm MeV}$. On the other hand, the mass splitting between $S$- and $P$-wave states with the same principle quantum number $n$ is sizable, e.g., the mass difference between $\eta_t(2S)$ and $h_t(1P)$ is $66~{\rm MeV}$.
• Figure 2: The $\dd\sigma(pp\to\eta_t)/\dd m_{t\bar{t}}$ distributions at the 13-TeV LHC calculated using the leading-order Green function method (blue) and the B-W formula with the first three $\eta_t (nS)$ states superposed (red). The two dashed vertical lines represent $m_{t\bar{t}}=M_{\eta_t}\pm\Gamma$ with $M_{\eta_t}=342.08~{\rm GeV}$ and $\Gamma=2.84~{\rm GeV}$, respectively, between which we identify the $t\bar{t}$ system to be on "resonance". We also show the individual $nS$ resonances with $n=1$ (red), $n=2$ (orange), and $n=3$ (yellow) in dashed curves.
• Figure 3: The theoretical contours on the $M_{\eta_t}$--$\sigma(pp\to\eta_t)$ (13-TeV LHC) plane derived from varying $\kappa\equiv \mu_R\times r$ with $m_t=172.4~{\rm GeV}$ (red), $m_t=172.4+0.7~{\rm GeV}$ (orange), and $m_t=172.4-0.7~{\rm GeV}$ (purple), respectively, according to the pole mass $m_t^{\rm pole}=172.4\pm0.7$ GeV reported in Ref. ParticleDataGroup:2024cfk. We also present the experimental results reported in Ref. CMS:2025kzt (blue) and Ref. ATLAS:2026dbe (green), respectively, at $1\sigma$ (thick lines) and 95% CL (thin lines) assuming $m_t=m_t^{\rm MC}=172.5$ GeV and $M_{\eta_t}=343$ GeV. We neglect the $K$-factor since it is close to one as discussed in Ref. Sumino:2010bv.
• Figure 4: The normalized $m_{t\bar{t}}$ distribution from reconstructed $t\bar{t}$ events for the $\eta_t b\bar{b}$ signal (red) and the $t \bar{t}b\bar{b}$ background (blue). The green dashed lines indicate the selection window $330~{\rm GeV}<m_{t\bar{t}}<370~{\rm GeV}$.
• Figure 5: Example Feynman diagrams of the specified processes considered in this study.